Question 1

The area bounded by $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 4 y$ is

Accepted Answer

A)  $\frac { 17 } { 3 }$ 
B)  $\frac { 16 } { 3 }$ 
C)  $\frac { 14 } { 3 }$ 
D)  $\frac { 13 } { 3 }$ 
E)  $\frac { 11 } { 3 }$ 
A)  $\frac { 17 } { 3 }$ 
B)  $\frac { 16 } { 3 }$ 
C)  $\frac { 14 } { 3 }$ 
D)  $\frac { 13 } { 3 }$ 
E)  $\frac { 11 } { 3 }$

Question 2

If R is the region bounded by $y = \sqrt { 4 - x ^ { 2 } } , y = x , y = - x$ then$\iint$R $\left( x ^ { 2 } + y ^ { 2 } \right) d A$  in polar form is

Accepted Answer

A)  $\int _ { \frac { \pi } { 2 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
B)  $\int _ { - \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
C)  $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
D)  $\int _ { - \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
E)  $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
A)  $\int _ { \frac { \pi } { 2 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
B)  $\int _ { - \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
C)  $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 2 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
D)  $\int _ { - \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$ 
E)  $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \left[ \int _ { 0 } ^ { 2 } r ^ { 3 } d r \right] d \theta$

Question 3

By Fubini's Theorem, the double integral$\iint$R $\sqrt { x y } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}$ is

Accepted Answer

A)  $\frac { 41 } { 9 }$ 
B)  $\frac { 29 } { 4 }$ 
C)  $\frac { 13 } { 9 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 4 } { 9 }$ 
A)  $\frac { 41 } { 9 }$ 
B)  $\frac { 29 } { 4 }$ 
C)  $\frac { 13 } { 9 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 4 } { 9 }$

Question 4

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $r = 2 + \cos ( \theta ) , 0 \leq \theta \leq \pi$ and the polar axis with area density $\rho ( r , \theta ) = \sin \theta$ is

Accepted Answer

A)  $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$ 
B)  $\left( 0 , \frac { 21 } { 10 } \right)$ 
C)  $\left( - \frac { 57 } { 44 } , 0 \right)$ 
D)  $\left( \frac { 42 } { 65 } , \frac { 19 \pi } { 52 } \right)$ 
E)  $\left( \frac { 8 } { 5 } , \frac { 8 } { 5 } \right)$ 
A)  $\left( \frac { 15 } { 28 } , \frac { 5 } { 8 } \right)$ 
B)  $\left( 0 , \frac { 21 } { 10 } \right)$ 
C)  $\left( - \frac { 57 } { 44 } , 0 \right)$ 
D)  $\left( \frac { 42 } { 65 } , \frac { 19 \pi } { 52 } \right)$ 
E)  $\left( \frac { 8 } { 5 } , \frac { 8 } { 5 } \right)$

Question 5

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $x = 3 , y = 2 , x = 0$ and the x-axis with area density $\rho ( x , y ) = x y ^ { 2 }$ about the y-axis is

Accepted Answer

A) 54 
B) 48 
C) 45 
D) 42 
E) 38 
A) 54 
B) 48 
C) 45 
D) 42 
E) 38

Question 6

The partial integral $\int _ { 1 } ^ { 2 } \frac { x } { y ^ { 2 } } d y$ is

Accepted Answer

A)  $\frac { x } { 42 }$ 
B)  $\frac { x } { 3 }$ 
C)  $\frac { x } { 2 }$ 
D)  $2 x$ 
E)  $3 x$ 
A)  $\frac { x } { 42 }$ 
B)  $\frac { x } { 3 }$ 
C)  $\frac { x } { 2 }$ 
D)  $2 x$ 
E)  $3 x$

Question 7

If $\sigma ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ is the volume density of the solid between two concentric spheres with radii 1 and 2, then its mass is

Accepted Answer

A)  $18 \pi$ 
B)  $15 \pi$ 
C)  $12 \pi$ 
D)  $8 \pi$ 
E)  $6 \pi$ 
A)  $18 \pi$ 
B)  $15 \pi$ 
C)  $12 \pi$ 
D)  $8 \pi$ 
E)  $6 \pi$

Question 8

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ lying within $x ^ { 2 } + y ^ { 2 } = 9$ is

Accepted Answer

A)  $\pi ( 144 - 72 \sqrt { 3 } )$ 
B)  $\pi ( 144 - 72 \sqrt { 5 } )$ 
C)  $\pi ( 144 - 71 \sqrt { 3 } )$ 
D)  $\pi ( 144 - 71 \sqrt { 5 } )$ 
E)  $\pi ( 144 - 70 \sqrt { 3 } )$ 
A)  $\pi ( 144 - 72 \sqrt { 3 } )$ 
B)  $\pi ( 144 - 72 \sqrt { 5 } )$ 
C)  $\pi ( 144 - 71 \sqrt { 3 } )$ 
D)  $\pi ( 144 - 71 \sqrt { 5 } )$ 
E)  $\pi ( 144 - 70 \sqrt { 3 } )$

Question 9

The cylindrical coordinates of the given rectangular coordinates (0, 1, -3) are

Accepted Answer

A)  $\left( 1 , \frac { \pi } { 6 } , - 3 \right)$ 
B)  $\left( 1 , \frac { \pi } { 4 } , - 3 \right)$ 
C)  $\left( 1 , \frac { \pi } { 3 } , - 3 \right)$ 
D)  $\left( 1 , \frac { \pi } { 2 } , - 3 \right)$ 
E)  $\left( - 1 , \frac { \pi } { 2 } , - 3 \right)$ 
A)  $\left( 1 , \frac { \pi } { 6 } , - 3 \right)$ 
B)  $\left( 1 , \frac { \pi } { 4 } , - 3 \right)$ 
C)  $\left( 1 , \frac { \pi } { 3 } , - 3 \right)$ 
D)  $\left( 1 , \frac { \pi } { 2 } , - 3 \right)$ 
E)  $\left( - 1 , \frac { \pi } { 2 } , - 3 \right)$

Question 10

If $\rho ( x , y , z ) = x$ is the volume density function of a solid enclosed by $100 x + 25 y + 16 z = 400$ and all the coordinate planes, its total mass is

Accepted Answer

A)  $\frac { 800 } { 3 }$ 
B)  $\frac { 800 } { 7 }$ 
C)  $\frac { 800 } { 9 }$ 
D)  $\frac { 800 } { 11 }$ 
E)  $\frac { 800 } { 13 }$ 
A)  $\frac { 800 } { 3 }$ 
B)  $\frac { 800 } { 7 }$ 
C)  $\frac { 800 } { 9 }$ 
D)  $\frac { 800 } { 11 }$ 
E)  $\frac { 800 } { 13 }$

Question 11

The surface area of the surface cut from $36 x + 16 y + 9 z = 144$ by x = 0, y = 0, and z = 0 is

Accepted Answer

A)  $6 \sqrt { 8 }$ 
B) 9 
C)  $15 \sqrt { 6 }$ 
D)  $\frac { 149 } { 15 }$ 
E)  $4 \sqrt { 1633 }$ 
A)  $6 \sqrt { 8 }$ 
B) 9 
C)  $15 \sqrt { 6 }$ 
D)  $\frac { 149 } { 15 }$ 
E)  $4 \sqrt { 1633 }$

Question 12

Let $x = r \cos \theta$ and $y = r \sin \theta$ Then $\frac { \partial ( x , y ) } { \partial ( r , \theta ) }$ is

Accepted Answer

A)  $2$ 
B)  $r ^ { 2 }$ 
C)  $r$ 
D)  $2 r ^ { 2 }$ 
E)  $\frac { r ^ { 2 } } { 2 }$ 
A)  $2$ 
B)  $r ^ { 2 }$ 
C)  $r$ 
D)  $2 r ^ { 2 }$ 
E)  $\frac { r ^ { 2 } } { 2 }$

Question 13

Using a change of variables, the double integral $\iint$R $( x + y ) ^ { 2 } \sin ^ { 2 } ( x - y ) d A$ where R is region bounded by the quadrilateral with vertices $( \pi , 0 ) , \left( \frac { 3 \pi } { 2 } , \frac { \pi } { 2 } \right) , ( \pi , \pi )$ and $\left( \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ is

Accepted Answer

A)  $- \frac { 7 \pi ^ { 4 } } { 12 }$ 
B)  $\frac { 7 \pi ^ { 4 } } { 12 }$ 
C)  $\frac { 5 \pi ^ { 4 } } { 12 }$ 
D)  $- \frac { 5 \pi ^ { 4 } } { 12 }$ 
E)  $- \frac { 11 \pi ^ { 4 } } { 12 }$ 
A)  $- \frac { 7 \pi ^ { 4 } } { 12 }$ 
B)  $\frac { 7 \pi ^ { 4 } } { 12 }$ 
C)  $\frac { 5 \pi ^ { 4 } } { 12 }$ 
D)  $- \frac { 5 \pi ^ { 4 } } { 12 }$ 
E)  $- \frac { 11 \pi ^ { 4 } } { 12 }$

Question 14

The iterated integral $\int _ { 0 } ^ { \frac { \pi } { 2 } }\int _ { 0 } ^  { \frac { \pi } { 2 } } \int _ { 0 } ^ { x z } \cos \left( \frac { y } { z } \right) d y d x d z$ is

Accepted Answer

A)  $\frac { \pi ^ { 2 } } { 2 }$ 
B)  $\frac { \pi ^ { 2 } } { 4 }$ 
C)  $\frac { \pi ^ { 2 } } { 8 }$ 
D)  $\pi ^ { 2 }$ 
E)  $2 \pi ^ { 2 }$ 
A)  $\frac { \pi ^ { 2 } } { 2 }$ 
B)  $\frac { \pi ^ { 2 } } { 4 }$ 
C)  $\frac { \pi ^ { 2 } } { 8 }$ 
D)  $\pi ^ { 2 }$ 
E)  $2 \pi ^ { 2 }$

Question 15

The surface area of the surface cut from $z = x + 2 y$ by x = 0, y = 0, x = 1, and y = 2 is

Accepted Answer

A)  $6 \sqrt { 8 }$ 
B)  $2 \sqrt { 6 }$ 
C)  $\sqrt { 6 }$ 
D) 18 
E)  $2 \sqrt { 1633 }$ 
A)  $6 \sqrt { 8 }$ 
B)  $2 \sqrt { 6 }$ 
C)  $\sqrt { 6 }$ 
D) 18 
E)  $2 \sqrt { 1633 }$

Question 16

The area bounded by $y = x ^ { 2 } - 9$ and $y = 9 - x ^ { 2 }$ is

Accepted Answer

A) 72 
B) 68 
C) 64 
D) 60 
E) 56 
A) 72 
B) 68 
C) 64 
D) 60 
E) 56

Question 17

The surface area of the surface $x ^ { 2 } + y ^ { 2 } = 3 z$ lying within $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 z$ is

Accepted Answer

A)  $\frac { \pi ( 7 \sqrt { 21 } - 5 ) } { 6 }$ 
B)  $\frac { \pi ( 7 \sqrt { 21 } - 7 ) } { 6 }$ 
C)  $\frac { \pi ( 7 \sqrt { 21 } - 9 ) } { 6 }$ 
D)  $\frac { \pi ( 7 \sqrt { 21 } - 11 ) } { 6 }$ 
E)  $\frac { \pi ( 7 \sqrt { 21 } - 13 ) } { 6 }$ 
A)  $\frac { \pi ( 7 \sqrt { 21 } - 5 ) } { 6 }$ 
B)  $\frac { \pi ( 7 \sqrt { 21 } - 7 ) } { 6 }$ 
C)  $\frac { \pi ( 7 \sqrt { 21 } - 9 ) } { 6 }$ 
D)  $\frac { \pi ( 7 \sqrt { 21 } - 11 ) } { 6 }$ 
E)  $\frac { \pi ( 7 \sqrt { 21 } - 13 ) } { 6 }$

Question 18

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x$ is switched, the result is

Accepted Answer

A)  $\int _ { 0 } ^ { 8 } \left[ \int _ { \sqrt [ 4 ] { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$ 
B)  $\int _ { 0 } ^ { 8 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 2 ] { y } } f ( x , y ) d x \right] d y$ 
C)  $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$ 
D)  $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$ 
E)  $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$ 
A)  $\int _ { 0 } ^ { 8 } \left[ \int _ { \sqrt [ 4 ] { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$ 
B)  $\int _ { 0 } ^ { 8 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 2 ] { y } } f ( x , y ) d x \right] d y$ 
C)  $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$ 
D)  $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { y } } ^ { \frac { y } { 4 } } f ( x , y ) d x \right] d y$ 
E)  $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { y } { 4 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$

Question 19

The surface area of the surface $x ^ { 2 } + z ^ { 2 } = 4$ lying within $x ^ { 2 } + y ^ { 2 } = 4$ is

Accepted Answer

A) 38 
B) 36 
C) 34 
D) 32 
E) 28 
A) 38 
B) 36 
C) 34 
D) 32 
E) 28

Question 20

The iterated integral $\int _ { 1 } ^ { 2 } \left[ \int _ { 0 } ^ { 2 x } x y ^ { 3 } d y \right] d x$ is

Accepted Answer

A) 56 
B) 48 
C) 42 
D) 36 
E) 32 
A) 56 
B) 48 
C) 42 
D) 36 
E) 32

The area bounded by $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 4 y$ is

If R is the region bounded by $y = \sqrt { 4 - x ^ { 2 } } , y = x , y = - x$ then $\iint$ _{R $\left( x ^ { 2 } + y ^ { 2 } \right) d A$} in polar form is

By Fubini's Theorem, the double integral $\iint$ R $\sqrt { x y } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}$ is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $r = 2 + \cos ( \theta ) , 0 \leq \theta \leq \pi$ and the polar axis with area density $\rho ( r , \theta ) = \sin \theta$ is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $x = 3 , y = 2 , x = 0$ and the x-axis with area density $\rho ( x , y ) = x y ^ { 2 }$ about the y-axis is

The partial integral $\int _ { 1 } ^ { 2 } \frac { x } { y ^ { 2 } } d y$ is

If $\sigma ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ is the volume density of the solid between two concentric spheres with radii 1 and 2, then its mass is

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ lying within $x ^ { 2 } + y ^ { 2 } = 9$ is

The cylindrical coordinates of the given rectangular coordinates (0, 1, -3) are

If $\rho ( x , y , z ) = x$ is the volume density function of a solid enclosed by $100 x + 25 y + 16 z = 400$ and all the coordinate planes, its total mass is

The surface area of the surface cut from $36 x + 16 y + 9 z = 144$ by x = 0, y = 0, and z = 0 is

Let $x = r \cos \theta$ and $y = r \sin \theta$ Then $\frac { \partial ( x , y ) } { \partial ( r , \theta ) }$ is

The iterated integral $\int _ { 0 } ^ { \frac { \pi } { 2 } }\int _ { 0 } ^ { \frac { \pi } { 2 } } \int _ { 0 } ^ { x z } \cos \left( \frac { y } { z } \right) d y d x d z$ is

The surface area of the surface cut from $z = x + 2 y$ by x = 0, y = 0, x = 1, and y = 2 is

The area bounded by $y = x ^ { 2 } - 9$ and $y = 9 - x ^ { 2 }$ is

The surface area of the surface $x ^ { 2 } + y ^ { 2 } = 3 z$ lying within $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 z$ is

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x$ is switched, the result is

The surface area of the surface $x ^ { 2 } + z ^ { 2 } = 4$ lying within $x ^ { 2 } + y ^ { 2 } = 4$ is

The iterated integral $\int _ { 1 } ^ { 2 } \left[ \int _ { 0 } ^ { 2 x } x y ^ { 3 } d y \right] d x$ is

Preparing for Calculus

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Exam 15: Multiple Integrals

The area bounded by y2=4xy ^ { 2 } = 4 xy2=4x and x2=4yx ^ { 2 } = 4 yx2=4y is

If R is the region bounded by y=4−x2,y=x,y=−xy = \sqrt { 4 - x ^ { 2 } } , y = x , y = - xy=4−x2​,y=x,y=−x then ∬\iint∬ R (x2+y2)dA\left( x ^ { 2 } + y ^ { 2 } \right) d A(x2+y2)dA in polar form is

By Fubini's Theorem, the double integral ∬\iint∬ R xydA\sqrt { x y } d Axy​dA with R={(x,y):0≤x≤1,0≤y≤1}R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}R={(x,y):0≤x≤1,0≤y≤1} is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by r=2+cos⁡(θ),0≤θ≤πr = 2 + \cos ( \theta ) , 0 \leq \theta \leq \pir=2+cos(θ),0≤θ≤π and the polar axis with area density ρ(r,θ)=sin⁡θ\rho ( r , \theta ) = \sin \thetaρ(r,θ)=sinθ is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by x=3,y=2,x=0x = 3 , y = 2 , x = 0x=3,y=2,x=0 and the x-axis with area density ρ(x,y)=xy2\rho ( x , y ) = x y ^ { 2 }ρ(x,y)=xy2 about the y-axis is

The partial integral ∫12xy2dy\int _ { 1 } ^ { 2 } \frac { x } { y ^ { 2 } } d y∫12​y2x​dy is

If σ(x,y,z)=x2+y2+z2\sigma ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }σ(x,y,z)=x2+y2+z2​ is the volume density of the solid between two concentric spheres with radii 1 and 2, then its mass is

The surface area of the surface x2+y2+z2=36x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36x2+y2+z2=36 lying within x2+y2=9x ^ { 2 } + y ^ { 2 } = 9x2+y2=9 is

The cylindrical coordinates of the given rectangular coordinates (0, 1, -3) are

If ρ(x,y,z)=x\rho ( x , y , z ) = xρ(x,y,z)=x is the volume density function of a solid enclosed by 100x+25y+16z=400100 x + 25 y + 16 z = 400100x+25y+16z=400 and all the coordinate planes, its total mass is

The surface area of the surface cut from 36x+16y+9z=14436 x + 16 y + 9 z = 14436x+16y+9z=144 by x = 0, y = 0, and z = 0 is

Let x=rcos⁡θx = r \cos \thetax=rcosθ and y=rsin⁡θy = r \sin \thetay=rsinθ Then ∂(x,y)∂(r,θ)\frac { \partial ( x , y ) } { \partial ( r , \theta ) }∂(r,θ)∂(x,y)​ is

The iterated integral ∫0π2∫0π2∫0xzcos⁡(yz)dydxdz\int _ { 0 } ^ { \frac { \pi } { 2 } }\int _ { 0 } ^ { \frac { \pi } { 2 } } \int _ { 0 } ^ { x z } \cos \left( \frac { y } { z } \right) d y d x d z∫02π​​∫02π​​∫0xz​cos(zy​)dydxdz is

The surface area of the surface cut from z=x+2yz = x + 2 yz=x+2y by x = 0, y = 0, x = 1, and y = 2 is

The area bounded by y=x2−9y = x ^ { 2 } - 9y=x2−9 and y=9−x2y = 9 - x ^ { 2 }y=9−x2 is

The surface area of the surface x2+y2=3zx ^ { 2 } + y ^ { 2 } = 3 zx2+y2=3z lying within x2+y2+z2=4zx ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 zx2+y2+z2=4z is

If the order of integration of ∫02[∫x24xf(x,y)dy]dx\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x∫02​[∫x24x​f(x,y)dy]dx is switched, the result is

The surface area of the surface x2+z2=4x ^ { 2 } + z ^ { 2 } = 4x2+z2=4 lying within x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4 is

The iterated integral ∫12[∫02xxy3dy]dx\int _ { 1 } ^ { 2 } \left[ \int _ { 0 } ^ { 2 x } x y ^ { 3 } d y \right] d x∫12​[∫02x​xy3dy]dx is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Vector Calculus

Differential Equations

Filters

The area bounded by $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 4 y$ is

If R is the region bounded by $y = \sqrt { 4 - x ^ { 2 } } , y = x , y = - x$ then $\iint$ _{R $\left( x ^ { 2 } + y ^ { 2 } \right) d A$} in polar form is

By Fubini's Theorem, the double integral $\iint$ R $\sqrt { x y } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}$ is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $r = 2 + \cos ( \theta ) , 0 \leq \theta \leq \pi$ and the polar axis with area density $\rho ( r , \theta ) = \sin \theta$ is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $x = 3 , y = 2 , x = 0$ and the x-axis with area density $\rho ( x , y ) = x y ^ { 2 }$ about the y-axis is

The partial integral $\int _ { 1 } ^ { 2 } \frac { x } { y ^ { 2 } } d y$ is

If $\sigma ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ is the volume density of the solid between two concentric spheres with radii 1 and 2, then its mass is

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ lying within $x ^ { 2 } + y ^ { 2 } = 9$ is

If $\rho ( x , y , z ) = x$ is the volume density function of a solid enclosed by $100 x + 25 y + 16 z = 400$ and all the coordinate planes, its total mass is

The surface area of the surface cut from $36 x + 16 y + 9 z = 144$ by x = 0, y = 0, and z = 0 is

Let $x = r \cos \theta$ and $y = r \sin \theta$ Then $\frac { \partial ( x , y ) } { \partial ( r , \theta ) }$ is

The iterated integral $\int _ { 0 } ^ { \frac { \pi } { 2 } }\int _ { 0 } ^ { \frac { \pi } { 2 } } \int _ { 0 } ^ { x z } \cos \left( \frac { y } { z } \right) d y d x d z$ is

The surface area of the surface cut from $z = x + 2 y$ by x = 0, y = 0, x = 1, and y = 2 is

The area bounded by $y = x ^ { 2 } - 9$ and $y = 9 - x ^ { 2 }$ is

The surface area of the surface $x ^ { 2 } + y ^ { 2 } = 3 z$ lying within $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 z$ is

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { x ^ { 2 } } ^ { 4 x } f ( x , y ) d y \right] d x$ is switched, the result is

The surface area of the surface $x ^ { 2 } + z ^ { 2 } = 4$ lying within $x ^ { 2 } + y ^ { 2 } = 4$ is

The iterated integral $\int _ { 1 } ^ { 2 } \left[ \int _ { 0 } ^ { 2 x } x y ^ { 3 } d y \right] d x$ is